The fruits of string theory: The Shape of Inner Space

Physicists and mathematicians embrace in this book by Shing-Tung Yau.

String theory has its scientific origins in the late 1970s and early 1980s, but it was propelled into the full view of the public in 2000 thanks to Brian Greene's readily accessible and scientifically accurate (if mathematically devoid) prose in The Elegant Universe. In the intervening decade, the basic ideas of string theory—that the Universe is potentially made up of little strings—has become fairly well known by members of the public. However, with no real possible experimental test to directly probe for the existence of these tiny strings, many within and outside the scientific community question the validity of a "theory of everything" that has no readily testable predictions.

One thing that, in my experience, constantly gets overlooked is that, in order to pursue the study of string theory, entire fields of mathematics have been revolutionized and brought back from what was the dust bin of mathematical curiosity. Their resurgence and renewed interest exist completely independently of whether string theory is right or if it gets left alongside Ptolemaic epicycles in the annals of scientific ideas.

The book starts with the beginnings of geometry and its influence on physics by first discussing the Platonic solids and how the ancient Greeks viewed them in relation to the physical properties of the real world. It then jumps forward a few thousand years, to where Yau briefly mentions his childhood in China and his budding interest in geometry. Since the book is not an autobiography, it quickly shifts back to the mathematics that Yau has spent his career studying, including topology, algebraic geometry, and geometric analysis—the tools that laid the foundations for his later work.

Yau doesn't hold the reader's hand in the book, and he expects the reader to have an intuition capable of understanding some non-obvious mathematical and geometric ideas. For instance, the product of two S1 manifolds (circles in a plane) is a rectangle rolled up on itself twice (think about it for a minute). The book gets down to the core of its focus when Yau and Nadis introduce the Calabi conjecture, the geometric problem that gave rise to Calabi-Yau manifolds. Calabi's question boils down to this:

Can a compact Kähler manifold with a vanishing first Chern class also have a Ricci-flat metric?

How this turned the world of physics on its head is hard to see at first, but Yau explains this conjecture in more physical terms. The question, while purely mathematical in nature, can be intimately tied to Einstein's field equations for general relativity. Viewed through the lens of general relativity, this question could be rephrased as, "Could there be gravity in our Universe even if space was totally devoid of matter?" As Yau puts it, if Calabi was correct, then curvature alone would make gravity possible, even in the absence of matter.

The next section of the book focuses on Yau's solution to the Calabi conjecture, and it raises an interesting point that highlights the difference between mathematicians and scientists/engineers. Yau's proof to the Calabi conjecture in 1976 (published in two papers in 1977 and 1978) showed that a manifold that satisfied Calabi's requirements (compact, Kähler, Ricci-flat) exists... nothing more. Nothing about the nature of it, what it looks like, or if a solution in the form of a realization can actually be computed. As an engineer this doesn't help me one bit, but to a mathematician, as Yau puts it, the fact that an answer exists is the answer. There is no need to go further. It reminds me of an old joke:

In three separate locked, airtight, rooms sit a physicist, an engineer, and a mathematician. In each room is a sealed box with a key that will allow escape. Upon learning of their predicament, all get to working on the problem at hand. The physicist works out the material properties and forces needed to exactly pry the box open with a small screwdriver, applies the minimal amount of force needed, gets the key, and leaves. The engineer takes the brute force approach: he smashes the box to pieces, picks up the key, opens the door, and leaves. The mathematician is not heard from for weeks, and upon going back to look for him, the engineer and physicist find him dead in the room. In front of him are pages upon pages of a mathematical proof. At the end it is written, "a solution exists."

It wasn't until eight years later, in 1984, that physicists started realizing the importance of manifolds such as those described by Calabi-Yau in the study of string theory. As Yau explains it, it was the need for supersymmetry that lead to the mathematical concept of holonomy, which in turn lead to Calabi-Yau manifolds. Once the connection was made, physicists started creating actual realizations of Calabi-Yau manifolds and studying the properties of these spaces and whether or not they could predict the properties of the universe we see around us.

The remainder of the book discusses how the mathematics has advanced thanks to work of physicists and how physics has advanced thanks to the work of mathematicians. While the book focuses heavily on the mathematics, it is only in one of the last chapters where Yau really hammers his key point home: no matter what happens with string theory, the mathematics developed to describe it will still be correct and will form the foundation for future generations of mathematicians and geometers. The Shape of Inner Space proves to be an interesting read to anyone with an interest not only in string theory, but in the mathematics that underlie this potential theory of everything.

So the intersection of two circles is a shape bordered by two arcs that touch at both ends. I've been trying to imagine how one could take a rectangle, "roll it up twice," and end up with that shape. No luck here.

So the intersection of two circles is a shape bordered by two arcs that touch at both ends. I've been trying to imagine how one could take a rectangle, "roll it up twice," and end up with that shape. No luck here.

Roll the rectangle into a tube, then remained is spoiler

Spoiler: show

then connect the end of the tube. The resulting torus is the intersection of two circles (one tracing out another)

Calibi's question is like the old saw: If a tree falls in the forest, does it make a sound if there's no one there to hear it? As far as I'm concerned, the answer is, "What tree?"

If space were totally devoid of matter, there would be no way to detect the presence of gravity because there would be no matter for it to act upon. If gravity cannot be detected, there is no empirical basis for asserting that it exists. What fun is that?

I think you mean that the product (not intersection) of two S^1 manifolds (not S^2 manifolds), which are circles, gives a torus. An alternative way to construct a torus is to roll a rectangle up into a cylinder, and then join the ends of the cylinder together.

Also, it's "Calabi", not "Calibi". Just thought you'd like to know. Thanks for the review.

The author is obviously entirely unfamiliar with the math involved in string theory, so it puzzles me why he would presume to be in the position to characterize it as living "in the dust bin," which indeed it does not. In addition, the author shouldn't have said "intersection," but cross-product of two circles (S^1 is just a fancy name for a circle, no need to name-drop and mention manifolds).

"For instance, the intersection of two S2 manifolds (circles in a plane) is a rectangle rolled up on itself twice (think about it for a minute)."

The first parenthetical is misleading. As archtop noted, the intersection of two circles in a plane is either a point, a circle, or a region bounded by two arcs. That's not what you're talking about here, but it's what the parenthetical makes it sound like.

As JoshuaRDavis points out, an S2 manifold is a space topologically equivalent to the surface of a sphere; an S1 manifold is a circle in a plane. And yes, I think you're talking about the product, not the intersection.

I think you mean that the product (not intersection) of two S^1 manifolds (not S^2 manifolds), which are circles, gives a torus. An alternative way to construct a torus is to roll a rectangle up into a cylinder, and then join the ends of the cylinder together.

You are entirely correct, sorry for the brain fart, it is fixed now.

Quote:

Also, it's "Calabi", not "Calibi". Just thought you'd like to know. Thanks for the review.

The author is obviously entirely unfamiliar with the math involved in string theory, so it puzzles me why he would presume to be in the position to characterize it as living "in the dust bin," which indeed it does not.

For example, it's closely related to the proof of the Poincare conjecture, a mathematically important result independently of physical applications.

I hope this didn't come across as "beating up the author"! It seemed important, because the author was saying that this was the sort of mathematical insight the reader would be required to grasp in order to make it through the book, but the mistakes in the description made the claim not only obscure but actually inaccurate. People who weren't already familiar enough with topology to see what he was trying to describe might have concluded that they weren't math-skilled enough to understand the claim, which could have repelled a lot of readers from the book.

For the sake of clarity, an S1 manifold is a one-dimensional space which is locally flat, but which wraps around. It's like Mario in the original Mario Brothers, walking straight across the screen, not stopping at the right edge but just continuing straight on and reappearing on the left. Topologically it's equivalent to a circle, but using strictly local measurements, there's no way of telling that the space is at all curved--it looks completely flat.

The product of two S1 manifolds is a two-dimensional space in which each dimension behaves like that, like the map in some old RPG games (some of the Phantasy Star or Final Fantasy games have this, for instance). The insight mentioned in the article involves seeing how this space can be characterized.

Spoiler: show

In particular, the insight is that this is topologically equivalent to a *TORUS*. Most people who haven't thought about it sort of assume it's equivalent to a sphere. One way of seeing that this isn't the case is to consider a sphere, and a point on its equator. From this point, a path going due north and a path going due east will both wrap eventually wrap all the way around the sphere and arrive back at the same point--but first they will intersect on the far side of the sphere. There's no point you can choose on the S1xS1 surface (or RPG map) where the same is true. Rather, if you printed out the map as a rectangle, you'd need to make the entire top edge match up with the entire bottom edge, resulting in a tube...and then you'd need to make the entire right edge (now a circle) match up with the entire left edge, requiring you to bend the tube into a loop...making a donut / inner tube / torus shape.

*I'm sorry, I can't figure out how to do the "spoiler" tag properly, and google isn't being helpful!**Got it! Uses square brackets rather than angles.

For the sake of clarity, an S1 manifold is a one-dimensional space which is locally flat, but which wraps around.

ah, that clears it right up

maybe somebody should define a topological product? ok so a topological space is, roughly, just a shape of some kind, for example a circle. Note that I don't mean it has to be something you could draw, it could be three dimensional, or even higher dimensional. So a solid cube is also a topological space.

With that out of the way, the product is pretty easy to define: For spaces X and Y, the product space X x Y is given by taking a copy of space X for every point in space Y, and gluing together all the copies so that each point is glued to the same point in its neighboring copies.

So if X is a three inch long line, and Y is a beach ball, you'll put a copy of a three inch line at every point on the beach ball. After doing the first few thousand, what you'll get looks like a porcupine. Keep doing it long enough though, and you won't be able to see the gaps between the lines any more. Eventually you just end up with a three inch thick beach ball.

If you were able to follow my vague description of how you glue the copies together, you should be able to convince yourself that X x Y = Y x X.

I think "piling on" might have been a better characterization, but I did feel like the author deserved a bit of a thrashing for making me feel stupid for failing to understand what turned out to be the author's wrong description.

Your comments are helpful and courteous. I apologize for my unintended slight.

In fact, I thought the reason why supersymmetry is desirable is because it maximizes the symmetries a point particle can have: it gives a loophole for the Coleman-Mandula no-go theorem on mixing external (spacetime) symmetries with internal symmetries. [ http://en.wikipedia.org/wiki/Coleman%E2 ... la_theorem ] And we know nature likes it laws (symmetries and symmetry breakings).

Then string theory is a natural way to implement supersymmetry, because the string degrees of freedom looks like external degrees of freedom on a string world-"membrane" (the space it sweeps as it goes).

alansky wrote:

If space were totally devoid of matter, there would be no way to detect the presence of gravity because there would be no matter for it to act upon. If gravity cannot be detected, there is no empirical basis for asserting that it exists. What fun is that?

But that is not how standard cosmology works. General relativity is a property of the cosmology, whether or not you have passed reheating and got matter. In fact, in standard cosmology spacetime and its curvature is observed before particles, as the space inflation happens in.

I think standard cosmology makes much more sense than "quantum theories of gravity" and "theories of everything".

We know that galaxy structures are seeded by primordial fluctuations that we seen in CMB* and in galaxy cluster structure formation. But we also seem to observe by SN1986A photons that spacetime is smooth on the Planck scale (whether it is Calabi-Yau wrapped with smaller or larger dimensions or not).

It makes so much more sense if inflation is the field that has the primordial fluctuations, since that is the only field we observe at that stage.

Of course, if something like eternal inflation is true we have pushed "blueshift" Planck energy singularities onto inflation instead, I think. But maybe it is much easier to take care of them in a scalar field theory than in gravity when it comes to cosmology. The scalar higgs field seems nice enough. Black holes seems tough enough even when they happen within spacetime!

-----------------------* Note that this is another way to observe that general relativity exists before particles exists. Inflation has blown up primordial fluctuations within spacetime, fluctuations that existed before particles!

The book sounds interesting, but I think I need one with more handholding.Maybe "The Shape of Inner Space For Dummies (tm)".

I have a shelf full of them, and my favourite is "The trouble with physics today" by Lee Smolin. Although it is slightly dated, in that the LHC is operation, the Higgs Boson has been discovered, and some of the theorised alternatives to string theory discussed in the book have since been disproved, it is still the best summary of the last 100+ years of physics I have read.

I particularly liked the rebuttal to those critics that string theory is untestable. Not only has it broadened mathematics, but there is one other key point: it is inexpensive to study. I don't understand what all the moaning is about when we are paying a few salaries and buying a few whiteboards.

Sitting here with my Pet Higgs Boson who tells me he's responsible for mass, henceforth gravity, I am reluctant to discuss with him the conjecture of gravity without mass. I gently asked my Higgs if "Curvitrons" could be discovered that provide gravity through membrane curvature in a perfectly aligned Calabi-Yau Manifold.He answered that we will need a super-symmetrical lead-aluminum (PbAl) collider operating at 60bTeV in nDimensional space (built in China) to even begin to approach the conjecture.I agreed to guy him a Mercedes if he would just shut-up.

I think you mean that the product (not intersection) of two S^1 manifolds (not S^2 manifolds), which are circles, gives a torus. An alternative way to construct a torus is to roll a rectangle up into a cylinder, and then join the ends of the cylinder together.

You are entirely correct, sorry for the brain fart, it is fixed now.

Quote:

Also, it's "Calabi", not "Calibi". Just thought you'd like to know. Thanks for the review.

I am going to kick my spell checker in the nuts. That's fixed too.

Great to see it is all fixed (and I now feel slightly less stupid for not being able to visualise the intersection turning into a cylinder) but as a professional journalist you really need to drop the adolescent "brain fart", "kick in the nuts" shtick. You will likely find that the vast majority of the Ars Technica paying subscribers have not been 13 for a long time. Aside from that, keep up the good work.

To physicists string theorists are simply rabble rousers,To engineers the string is queer - (as it won't hold up their trousers),A geometer can cope better, but a Buddist knows what's what:Just look inside and you will see, the Universe is Knot!

I have a shelf full of them, and my favourite is "The trouble with physics today" by Lee Smolin. Although it is slightly dated, in that the LHC is operation, the Higgs Boson has been discovered, and some of the theorised alternatives to string theory discussed in the book have since been disproved, it is still the best summary of the last 100+ years of physics I have read.

That book has been universally panned by physicists what I know of.

And no wonder, Smolin is one of a minor group of mathematicians at the core, a priori supporting such an alternative. The problem with his chosen physics is two-fold, generally axiomatic methods doesn't work in physics (say, quantization of field theories has never been axiomatized) and specifically Loop Quantum Gravity neither obey relativity nor can display dynamics (can't predict a simple harmonic oscillator to build dynamics out of). It is a non-starter.

So I wouldn't expect his book to be especially faithful to physics and its history.

@ elhombre:

I thought it was a refreshing variant! Anything is possible in love and text making, as long as you don't need to be faithful. (In the latter case, to the specific content. In the former case, it's moral relativistic - can't help you there. (O.o))

Of course, if it becomes a habit it would eventually be boring. But has it?

I got the Kindle version of this book and I have to admit that it's the best laid out maths book that I have on Kindle. (Maths books on Kindle always seem to have a million errors in them and all manner of layout issues - too hard to proof read, I guess.)

But I just loved this book for the simple reason being that it didn't water the good stuff down. It expects you to be able to follow a discussion in modern maths. It was a joy reading a science/maths book that had lots of detail in it and didn't feel so simplified that it felt like it was skipping important stuff. I wish more popular science/maths books tried to be of this level and quality.

The only minor problem that I did have was the book didn't really feel like it had a theme or guiding direction. Although the author avoided putting too much autobiographical material in the book, the structure did come across as being a set of work that Yau studied through his life. Nothing wrong with that as what Yau studied was wonderful stuff and I loved hearing about it.

I know this is just a silly post, but I'm in a pedantic mood (sorry) for some reason so...

This (presently) isn't true. The Higgs field is responsible for inertial mass. Gravitational mass is (at this point at least) a different thing. They are proportional. The Higgs field and boson are part of the Standard Model which does not incorporate General Relativity (gravity) - in fact General Relativity and the Standard Model are incompatible. Also, the Higgs field is only responsible for the mass of some things, the vast majority of the mass of "ordinary stuff" does not result from Higgs.

You are entirely correct, sorry for the brain fart, it is fixed now.I am going to kick my spell checker in the nuts. That's fixed too.

Great to see it is all fixed (and I now feel slightly less stupid for not being able to visualise the intersection turning into a cylinder) but as a professional journalist you really need to drop the adolescent "brain fart", "kick in the nuts" shtick. You will likely find that the vast majority of the Ars Technica paying subscribers have not been 13 for a long time. Aside from that, keep up the good work.

Oh come on, just because you don't like it doesn't mean that other people don't find it amusing. As long as it doesn't come through in the articles, and it's limited to abuse of abstract things in the comments, then I don't see the problem.

"Professionalism" tends to be a euphemism for "politically correct", and I don't really want a politically correct science writer.

The Higgs field is responsible for inertial mass. Gravitational mass is (at this point at least) a different thing. They are proportional. The Higgs field and boson are part of the Standard Model which does not incorporate General Relativity (gravity) - in fact General Relativity and the Standard Model are incompatible. Also, the Higgs field is only responsible for the mass of some things, the vast majority of the mass of "ordinary stuff" does not result from Higgs.

Pedantry on the pedantic. =D This is mostly correct, and I believe you have to be a particle physicist to see why the Higgs field couples mostly linearly. (Not so with photons, gluons, neutrinos, and apparently not linearly with its free boson. The Z & Ws not included, since they are composite* with some of the higgs field's bosons.)

But I don't think GR & SM are incompatible. SM is compatible with special relativity and is a quantizable gauge theory I think. GR is compatible with special relativity (well, duh) and is a gauge theory I think. It too is quantizable by way of its Lagrangian, as expected. The graviton that is thus predicted is compatible with the SM in the sense that as a spin 2 fundamental particle it doesn't conflict with any of the SM fundamental particles.

The problem is that neither GR nor SM are fundamental theories but known to be effective theories.

In the case of GR it shows up when it, and its quantization, breaks down at high energies. GR displays singularities and the quantization becomes meaningless by non-convergence. That isn't supposed to happen in quantum field theories, I think. Blah blah non-renormalizable blah.

In the case of SM it shows up as an enormous finetuning. I don't know if it is true, but I once saw the claim that if you add all SM finetunings up, it is as severely finetuned as the cosmological constant (a factor of ~ 10^120)! And probably for the same reason, I should think, while GR is coupled to cosmology so aren't SM and its vacuum CC predictions as of yet. Perhaps a discovery of supersymmetry will start to edge particle physics towards that.

If you want to put a fine point to it, eventually SM with higgs breaks down too. If the 125-126 GeV higgs is a standard higgs it predicts a quasistable vacuum, currently @ 2 sigma. (LHC needs to finesse SM parameters, maybe they can tell at 3 sigma sometime soon.)

So SM has a sort-of singularity "in time". Which is curious come to think about it, time is a conjugate pair with energy as in Noether's theorems I take it. So GR breaks down at high energies, while SM w higgs breaks down at "low energies" (long times).

This is, by the way, in a handwavy way a global problem in the sense that energy is a global property and a global symmetry in Noether's theorems (I take it), a "charge" conserved but not by local symmetries as particle charges are. When energy measures breaks down because the system description does, if so locally, the physics of the global system is inconsistent. (Again, well, duh.) Another connection with cosmology perhaps.

So for all practical concerns GR & SM are compatible and sort of akin, but rather separate. And evidently we need to marry cosmology (GR) with particle physics (SM) to make progress on both.

In the context I note that string theory, while being a good frame (theories for new particles, landscape for cosmologies), isn't strictly necessary for doing some of that.

* I don't think that is the technical term though. Hadrons like protons are composite in another and more dynamical sense, composed by dynamically interacting quarks and gluons. Z&Ws are more like chimeras.

elhombre wrote:

I didn't say "all", I saod only those over 13 years old.

That means all readers over 13 years and fewer below, which is stereotyped for individuals of both groups.

[Disclaimer: I happened to like Ford's humor. And I don't think stereotyping is always useful.]

In certain mathematical philosophies proving that something exists requires you to be able to show how it is built. Just proving that something can't not exist is seen as so much hocus pocus. To prove that something exists you aways have to show how to build it.

Stereotyping is very important ! It lets you dislike whole groups of people without having to go to the trouble of getting to know them first. It is very efficient.

When we seem to agree. Efficient when applicable, which is not always.

For example, I can dislike people (no one here, I rush to say) without stereotyping them. *That* is efficiency for ya'! =D

[ By the way, where is the instructions for coding in the editor? Trying this: [img src="http://arstechnica.com/civis/images/smilies/smile.png" alt=":)" title="Smile"], this: <img src="http://arstechnica.com/civis/images/smilies/smile.png" alt=":)" title="Smile"> , and C&P: .

In certain mathematical philosophies proving that something exists requires you to be able to show how it is built. Just proving that something can't not exist is seen as so much hocus pocus. To prove that something exists you aways have to show how to build it.

- As you say, constructivism is a philosophy. Not a very good one either, because using idealization (say, infinity instead of unboundedness) works better in both math and physics as the results covers a larger space.

Besides, on a physical level realism is built into both classical and quantum mechanics, which is what string theory lives in. Action-reaction and observation-observables embodies the old Johnson maxim of "when I hit a stone, it hits back" or in other words constrained reaction on constrained action. Anything observable with these theories exists by such a definition (when the observations and the theories constraining it shakes out so competitors are safely eliminated).

You may or may not have a problem with what it means to "exist" if it is an emergent phenomena. A table isn't a solid, most of it is vacuum between atoms. It only looks and behaves like a solid in an idealized sense. But it doesn't devalue the observable definition.

- String theory has an embarrassment of constructions. That is why the landscape, while being finite, is simply too large to efficiently cut down by math alone (I believe). Cosmology may help there.

Geometry to this level I really struggle with. Part if my poor brain is always trying to visualise things, while another part is pointing out that it's often impossible to do that, and that it only makes sense through the maths... which I don't know enough of

However, articles like this are food for thought, and generally lead me on a trawl through Wikipedia and the like in order to learn more about the subject. Thanks Ars, for keeping my grey matter ticking over.

elhombre wrote:

...as a professional journalist you really need to drop the adolescent "brain fart", "kick in the nuts" shtick. You will likely find that the vast majority of the Ars Technica paying subscribers have not been 13 for a long time. Aside from that, keep up the good work.

I'm 35, and heartily endorse the administration of virtual nut-kicking to intangible objects of frustration.

Of course, I'm not a paying subscriber (sorry Ars!) so maybe I just don't count.

So the intersection of two circles is a shape bordered by two arcs that touch at both ends. I've been trying to imagine how one could take a rectangle, "roll it up twice," and end up with that shape. No luck here.

Roll the rectangle into a tube, then remained is spoiler

Spoiler: show

then connect the end of the tube. The resulting torus is the intersection of two circles (one tracing out another)

maybe my problem with this is language and not the geometry. Since I learned most of my math (as little as it is) and science in spanish, I find that sometimes the words used in english, even if they are normally familiar to me, don't make sense. Can you instead show a graphic of this? I just am unable to use the words you have used to create a picture.

I'm 35, and heartily endorse the administration of virtual nut-kicking to intangible objects of frustration.

Of course, I'm not a paying subscriber (sorry Ars!) so maybe I just don't count.[/quote]

You are a potential subscriber so of course you count!

If the article had been about "Xtreme body piercing to the Max", or "10 hot tips for your orgasm workout" then the adolescent genitalia references would have been appropriate, but with string theory it is just jarring .. in my humble opinion.

When "nuts" and "fart" came out of an educated individual people would always say this, "Wow, you're cool, way to go, man!" But when it came out of a homeless drug addict low-life and has not taken a shower in years individual it's always sounds as disgusting as he looks. "You're disgusting, get a life, asshole!"

<sighs..>

Don't get me wrong, love to read about "nuts" and "fart". :-)

<Never thought I would commented on this science thread. But here I am. I've found a hole for me to get in.>

I guess in an ideal (topological?) world, rolling up a rectangle and then connecting the ends into a torus works, but I just keep seeing a scrunched-up inner ring and stretched/torn outer ring, since you can't take the same amount of matter and (easily) make it fold nicely into two difference circumferences..

(i.e. - try it with newspaper, and see if you can get anything remotely smooth... guess I'm just an engineer :-)

Matt Ford / Matt is a contributing writer at Ars Technica, focusing on physics, astronomy, chemistry, mathematics, and engineering. When he's not writing, he works on realtime models of large-scale engineering systems.